Bayes' theorem as two rectangles

• 0.3 and 0.7 are equally far from a fair coin ×1
• a coin with bias 0.6 reaches twenty to one almost always ×1
• a halfer told it is the first day moves above one half ×1
• a p-value of 0.05 is worth at most about 2.46 to one ×1
• a positive result is evidence for and a negative against ×1
• a prior with thinner tails has a finite expected amount ×1
• a thirder told it is the first day comes back to one half ×1
• adding decibels is multiplying odds ×1
• and at seven days it is 13/27 ×1
• and at three doors it is two thirds ×1
• and it is above a half for every pair ×1
• and the boy–boy block holds 2d − 1 of them ×1
• at a hundred million tosses even odds need about four standard errors ×1
• at a trillion tosses the Bayes boundary sits more than two standard errors above the iterated-logarithm envelope ×1
• at every middle amount the other envelope is worth 5/4 as much on average ×1
• at one awakening every account agrees on one half ×1
• at p = 0.05 and a million tosses the fair coin is favoured by more than a hundred to one ×1
• at p = 0.05 even the best single bias makes the result only about 6.8 times likelier than a fair coin ×1
• at p = 0.05 the uniform alternative and the fair coin are at even odds somewhere between 50 and 100 tosses ×1
• at the same z-score the best bias beats a fair coin by the same factor, about 6.8, at every sample size ×1
• at η near (1 − ρ)/(1 + ρ) the coverage is back to 95% ×1
• averaged over what might be seen, switching gains exactly nothing ×1
• Broome's partial sums grow without bound ×1
• combining p-values while choosing whether to continue inflates the error well past five per cent ×1
• each test alone keeps its false-positive rate ×1
• every order of the evidence ends at the same odds ×1
• every run ends almost certain of the bias 0.6, which is false ×1
• everybody has it, so everybody positive has it ×1
• fewer than one fair coin in twenty ever shows twenty to one against fairness ×1
• however the spread is chosen, p = 0.05 favours the biased coin by at most about two to one ×1
• Jahn's 104,490,000 trials: p below 0.0005 and the fair coin favoured about twelve to one ×1
• no fixed bet grows faster than the true likelihood ratio ×1
• nobody has it, so nobody who tests positive has it ×1
• one of the two strategies wins, and only one ×1
• one ticket a run is fair at one half ×1
• one ticket an awakening is fair at 1/(wakes + 1) ×1
• one to five positive likelihood ratios ×1
• prior is a probability, between 0 and 1 ×1
• seeing 1, the other is surely 2 ×1
• seeing any larger amount, the other is double with probability 2/5 ×1
• so its expected value is 1.1 times the amount seen ×1
• some fair coin leaves the 5% band at least four times ×1
• specificity is a probability, between 0 and 1 ×1
• staying wins as often as the first pick was right ×1
• switching gets better with more doors ×1
• switching wins two thirds against a host who knows, and half against one who does not ×1
• tails brings between 1 and 8 awakenings ×1
• the answer falls as the detail becomes commoner ×1
• the answer rises with the base rate ×1
• the area under the likelihood ratio is the Bayes factor for the biased coin ×1
• the asked families settle near 1/3 ×1
• the average over a uniform bias is many times smaller at the larger sample ×1
• the awakenings settle near one share in wakes + 1 ×1
• the bound is the best calibrator ×1
• the calibration −1/(e p ln p) puts p = 0.05 at about 2.5 to one ×1
• the chance of the larger is a half plus half the chance the threshold falls between the two ×1
• the closest hypothesis in Kullback–Leibler divergence is 0.6 ×1
• the count is the formula (2 − p)/(4 − p) ×1
• the coverage is what the inflated variance predicts ×1
• the curve agrees with the grid count ×1
• the error rate is near Wald's bound 1/(1 + 10^(threshold/10)) ×1
• the formula is the kept area ×1
• the four cells fill the square ×1
• the game needs at least three doors ×1
• the halfer's square gives heads one half ×1
• the ignorant host's story throws two of the six worlds away ×1
• the ignorant-host variant is only drawn at three doors, where switching has one meaning ×1
• the iterated-logarithm envelope has passed 1.96 ×1
• the kept area gives the protocol's answer ×1
• the largest pair is 2^N and 2^(N+1), N between 2 and 10 ×1
• the learned product passes 20, and a fresh prior in every study falls behind it ×1
• the marked row and column share one cell ×1
• the met-a-child families settle near 1/2 ×1
• the mixture's rate rises towards the best rate ×1
• the more the errors are shared, the less the second test adds ×1
• the only news drawn is that it is the first day ×1
• the paths differ on the way ×1
• the product of e-values passes 20 at most one time in twenty, whatever the continuation rule ×1
• the product of learned e-values is the Bayes factor of all the tosses pooled ×1
• the question is any, older or met ×1
• the runs settle near one half ×1
• the same coin's Bayes factor never reaches twenty to one ×1
• the shaded fraction is Bayes' theorem ×1
• the share of fair coins whose Bayes factor ever reaches k stays below 1/k ×1
• the single bet beats the uniform prior near its own guess and is blind to a coin that favours tails ×1
• the spread that suits the biased coin best shrinks with the sample ×1
• the thirder's square gives heads one share in wakes + 1 ×1
• the trait takes between 1 and 12 values ×1
• the two curves are mirror images about one half ×1
• the two strategies exhaust the possibilities ×1
• the typical Bayes factor falls steadily as tosses accumulate ×1
• the typical stopping point is of the same order as the fixed sample a 5% test with 80% power needs ×1
• the view is one of sweep, monty, monty-scale, children, tuesday, trait, mention, children-sim, beauty, beauty-sim, beauty-bets, beauty-many, oddsladder, oddsorders, oddsdependent, oddssequential, oddsworth, envbounded, envbroome, envexpect, envsim, envthreshold, misclosest, miskl, misdrift, mistie, miscover, mistemper, lindhero, lindlike, lindbound, lindthresh, lindprior, oshero, ospath, osville, osmean, osboundary, ospower, evhero, evcont, evgrow, evcost, evconvert, evprior ×1
• the walk's slope is the difference of the two divergences ×1
• the worlds fill the square ×1
• the z-score for even odds is close to √ln(2n/π) ×1
• under a fair coin the Bayes factor for a coin of unknown bias averages exactly one at every number of tosses ×1
• watched for 200 tosses, a fair coin passes 20 at most one time in twenty ×1
• watching after every toss, more than half of fair coins reach p < 0.05 within 100,000 tosses ×1
• watching for twenty to one needs more tosses than a single test at five per cent ×1
• with no shared cause the product is exact ×1
• with the number of studies fixed in advance, Fisher's method holds its level ×1

The whole library · What the figures prove